Nonstationary equations for the reaction layer with the degenerate equilibrium points

A. A. Bykov ; K. E. Ermakova

Modelirovanie i analiz informacionnyh sistem, Tome 24 (2017) no. 3, pp. 309-321.

Résumé

We consider a nonstationary process of spreading some substance in a one-dimensional spatially inhomogeneous system of cells. It is assumed that a change in the concentration of $u_n(t)$ in a cell with the number $n$ with time $t$ is determined by the difference in concentration in this cell and in its two neighbors on the left and on the right, as well as the source density, which depends on $n$ and depends on $u_n(t)$. Such a model leads to the initial-boundary value problem for the differential-difference equation (differentiation with respect to $t$ variable, the difference expression with respect to $n$). With a sufficiently small difference in concentration in each pair of neighboring cells we can replace the difference expression by the second partial derivative with respect to the spatial coordinate, and describe the propagation by the reaction-diffusion equation. This equation belongs to the class of quasilinear parabolic equations. It is assumed that the density of the sources vanishes (with changing the sign) at three values of the concentration, two of which, lower and upper, are stable. There is also an intermediate unstable state with zero source density, in which the sign reversal also takes place. The peculiarity of our model is that we assume, that two extreme roots of the source density function are degenerate (with an integer or fractional exponent). We intend to show analytically and by the computer simulation, that this model leads to the fact, that the rate of asymptotic aspiration of concentration to equilibrium values for a moving front becomes power-law instead of exponential, which takes place for standard models. In the paper, we have constructed a formal asymptotics solution of the initial-boundary value problem for the reaction-diffusion equation in a homogeneous medium with a power-law dependence of the source density on the temperature, an upper and lower solutions are constructed, a rigorous justification of the formal asymptotics is given. Precise solutions of the diffusion reaction equation are constructed for a wide class of source density functions.

Détail

Keywords: nonlinear differential equations, asymptotic methods, contrast structures, differential inequalities
Mots-clés : combustion.

Bibliographie

[1] Butuzov V. F., “On Periodic Solutions to Singularly Perturbed Parabolic Problems in the Case of Multiple Roots of the Degenerate Equation”, Comp. Math. Math. Phys., 51:1 (2011), 40–50 | DOI | MR | Zbl

[2] Butuzov V. F. et al., “On a singularly perturbed initial value problem in the case of a double root of the degenerate equation”, Nonlinear Anal. Theory, Meth. and Appl., 83 (2013), 1–11 | DOI | MR | Zbl

[3] Butuzov V. F., “On the Special Properties of the Boundary Layer in Singularly Perturbed Problems with Multiple Root of the Degenerate Equation”, Math. Notes, 94:1 (2013), 60–70 | DOI | DOI | MR | Zbl

[4] Alshin A. B., et al., “The traveling wave as a solution of a nonlinear equation in semiconductors with strong spatial dispersion”, Comp. Math. Math Phys., 48:5 (2008), 764–768 | DOI | MR | Zbl

[5] Butuzov V. F., Nefedov N. N., Schneider K. R., “Singularly perturbed problems in case of exchange of stabilities”, J. Math. Sci., 121:1 (2004), 1973–2079 | DOI | MR | Zbl

[6] Vasil'eva A., Nikitin A., Petrov A., “Stability of contrasting solutions of nonlinear hydromagnetic dynamo equations and magnetic fields reversals in galaxies”, Geophysical and Astrophysical Fluid Dynamics, 78:1–4 (1994), 261–279 | DOI | MR

[7] Bykov A. et al., “Anomalous Persistence of bisymmetric Magnetic Structures in Spiral Galaxies”, MNRAS, 292:1 (1997), 1–10 | DOI

[8] Moss D., Petrov A., Sokoloff D. et al., “The motion of magnetic fronts in spiral galaxies”, Geophysical and Astrophysical Fluid Dynamics, 92:1–2 (2000), 129–149 | DOI | MR

[9] Bozhevol’nov Yu. V., Nefedov N. N., “Front motion in the parabolic reaction-diffusion problem”, Comp. Math. Math. Phys., 50:2 (2010), 264–273 | DOI | MR | Zbl

[10] Pao C. V., Nonlinear parabolic and elliptic equations, Plenum, New York, 1992 | MR | Zbl